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Kinematic Dynamo Theory: The Dyson Equation and the Large‐Scale Field; the Bethe‐Salpeter Equation and the Fluctuation Intensity

### Abstract

An exact statistical set of kinematic dynamo equations is given representing turbulent generation of both a large‐scale magnetic field and a small‐scale turbulent field. These equations do not rely on approximately solving the fluctuation equations and using the results in the ordered field equations as do most treatments of statistical kinematic dynamos. Instead they treat both the fluctuation equations and the ordered field equation exactly. The results obtained indicate several points. First: The fluctuation intensity equation has the character of the Bethe‐Salpeter equation. Depending on whether one uses the long‐slow or the short‐sudden approximation there may, or may not, be an upper limit on the velocity turbulence in order that the energy density stored in the magnetic fieldfluctuations remain finite. No such restriction is found using approximate kinematic dynamo equations. Second: The normal modes of the large‐scale field (which obeys a Dyson equation) may, or may not, be mirrored in the singular eigenmodes of the fluctuation intensity equation. Third: the structure of the exact statistical kinematic dynamo equations is very different from the structure of the approximate kinematic dynamo equations‐particularly in the equation describing the fluctuation intensity. We have done this problem in order to demonstrate that the exact solution of at least one problem in statistical kinematicdynamo theory introduces new and interesting phenomena which are not brought to light in approximate treatments of the same phenomena.

© 1971 The American Institute of Physics

Received Tue Feb 16 00:00:00 UTC 1971
Published online Tue Oct 28 16:17:14 UTC 2003